# Two-sample test for multivariate normal distributions under the assumption that means are the same

Let $\{x_i\}_{i=1}^n$ be a sample from a multivariate Gaussian distribution ${\cal N}(0, \Sigma_X)$ and $\{y_i\}_{i=1}^m$ be a sample from ${\cal N}(0, \Sigma_Y)$.

Are there hypothesis tests for $\Sigma_X = \Sigma_Y$? Pointers to relevant literature would be very appreciated.

The Mauchly's test allows to test if a given covariance matrix is proportional to a reference (identity or other) and is available through mauchly.test() under R. It is mostly used in repeated-measures design (to test (1) if the dependent variable VC matrices are equal or homogeneous, and (2) whether the correlations between the levels of the within-subjects variable are comparable--altogether, this is known as the sphericity assumption).